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Eigenvalues of a Matrix

Source: VTRMC 2022 P5

January 4, 2023
VTRMCcollege contestslinear algebramatrixeigenvalue

Problem Statement

Let AA be an invertible n×nn \times n matrix with complex entries. Suppose that for each positive integer mm, there exists a positive integer kmk_m and an n×nn \times n invertible matrix BmB_m such that Akmm=BmABm−1A^{k_m m} = B_m A B_m ^{-1}. Show that all eigenvalues of AA are equal to 11.
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